Here we show performance comparisons of the proposed method relative to four baseline methods for estimating the underlying covariance and inverse covariance matrices. i) Empirical covariance, which is the sample covariance. ii) The Ledoit-Wolf estimator [1]}, which uses shrinkage regularization to perform MAP estimation for the covariance matrix by assigning an inverse Wishart prior on the covariance matrix. iii) Gaussian Copula GraphicalLasso [2]}, which penalizes the covariance matrix in a different way, particularly, with L1-norm constraints on the precision matrix after transforming the data by using Gaussian copulas. Regularization forces the entries of the precision matrix to be sparse. iv) Factor Analysis [3]} uses another form of regularization of the covariance matrix by imposing low-rank structure. Each of these baseline methods is expected to perform best when the data, or its transformed version, is normally distributed.
For the Ledoit-Wolf, GraphicalLasso, and FA, as is customary, we first normalize the data by subtracting the mean and dividing by the variance, before running these methods. On the other hand, as the non-Gaussian counting nature of the data is explicitly modeled in our proposed model, our algorithm is run on the raw observations. For model selection in both FA and proposed model, we use the exact rank of the simulated dataset. The regularization coefficient of Gaussian Copula GraphicalLasso algorithm is estimated by using 5-fold cross-validation. For the Ledoit-Wolf algorithm, we used the expression for the shrinkage coefficient given in [1]}. We use scikit-learn implementations of the baseline methods.
